On Feb 16, 10:59 pm, fom <fomJ...@nyms.net> wrote:> On 2/16/2013 5:55 AM, Dan wrote:>> > I can't concentrate enough to understand the whole post, however :> > Machines can't decide whether 1.0... = 0.9999999... .>> No. But, people do. It is taken as a trivial "decision".>> The beginning of the post states that it was an exercise> in logic. In that respect, there was no meaning attached> to 1.000... = 0.999... .>> In the logical construction of the real numbers system> using Dedekind cuts, one must fix a choice as to which> kind of rational cut will be taken as canonical. The> cuts corresponding to the rational numbers will either> all be chosen relative to least upper bounds or they> will all be chosen relative to greatest lower bounds.>> The reason for this is that the identity of real numbers> relative to the construction is based upon the order> relation inherited from the rational numbers used in the> preceding logical construction.>> When one looks at this presentation in books on real analysis,> the general tendency is to fix the rational cuts relative> to least upper bounds. This comes from the tendency to view> the construction relative to initial sequences. But, when> one considers the canonical name we use because of the fact> that some ratios form quotients that halt relative to the> Euclidean long division, the canonical name for the rational> cuts should be oriented from fixing the rational cuts> relative to greatest lower bounds.>> > In general , machines can't decide the equality or inequality of real> > numbers ,or infinite strings in general ,without the 0.(9) = 1> > equivalence of real numbers.> > This comes as a consequence that all computable functions are> > continuous ,while equality is not.>> Nice remark. I have your link up in my browser and look forward> to reading it.>> I just completed a discussion of identity>> news://news.giganews.com:119/88qdnU1ZNsB9j4LMnZ2dnUVZ_uudn...@giganews.com>> or look for "when indecomposability is decomposable" on> sci.math/sci.logic>> Relative to the trichotomy of real numbers in relation to> Dedekind cuts, the simplest view of what you are> expressing is the quotient topology for a map from the> real numbers into a three point set. The example from> Munkres "Topology" goes something like:>> p(x)=a if x>0> p(x)=b if x<0> p(x)=c if x=0>> The quotient topology induced by this map is given by>> {{a},{b},{a,b},{a,b,c}}>> This example shows why the Dedekind cuts suffice to> construct the real numbers, but they do not reflect> the topology above because the logical construction> requires that rationals correspond with a uniform> choice with regard to the order. In fact, that choice> must be uniform with respect to the ordinal sequencing> of natural numbers as that order is what is held invariant> in a full-blown construction. So, a function like the> one above for a Dedekind cut rational has the form>> p(x)=a if x>=0> p(x)=b if x<0>> Moreover, this is the choice is precisely the direction> we make with respect to the trivial "decision" concerning> canonical representation associated with>> 1.0... = 0.9999999...>> To address this problem, one must look at the metrization> of pseudometrics. In "Topology" by Kelley, there is a> somewhat strange circumstance in the proof of the metrization> lemma in the chapter on uniform spaces. It seems that> Dedekind cuts are logically prior to Cantorian fundamental> sequences because metrization invokes the least upper bound> property in its proof.>> Now, what is important about a pseudometric is that the> relevant axiom is attaching a metric structure to an identity> relation in the logical sense. That is,>> x=y -> d(x,y)=0>> For a metric that axiom is>> x=y <-> d(x,y)=0>> So, in the hierarchy of logical definition, one obtains the> real numbers from Dedekind cuts relative to a logical identity> relation. Then, a definition of least upper bound and greatest> lower bound for that system may be defined. Then, provided that> the nature of relations used in the metrization lemma are> satisfiable, one uses the function constructed in that> proof to put a metric on the system of Dedekind cuts.>> Given any metric structure, the system conforms to the> definition of a metric space. Then, one can use the order> relation of the "new" rational numbers (those that correspond> with rational cuts) to define Cantorian fundamental sequences.>> It is at that point that the trichotomy expressed by>> p(x)=a if x>0> p(x)=b if x<0> p(x)=c if x=0>> is admitted.>> If you look in my other post, however, it is clear that> one could use the "real numbers" of a near-field on 9> symbols as the a, b, c in the example above.>> p(x)=1 if x>0> p(x)=-1 if x<0> p(x)=0 if x=0>> Or, looking at the canonical order for my labels in> that post, and treating a finite subsequence circularly,>> p(x)=NAND if x>0> p(x)=NOR if x<0> p(x)=IMP if x=0>> That canonical order is given as>> <LEQ, OR, DENY, FLIP, NIF, NTRU, AND, NIMP, XOR, IMP, NAND, TRU, IF,> FIX, LET, NOR>>> The subsequence in which I am interested is>> <IMP, NAND, TRU, IF, FIX, LET, NOR>>> And, since NOR is the last in the sequence, the order of the> three elements I want relative to wrapping the subsequence> is given by>> <NOR, IMP, NAND>>> In any case, the point is that your observation speaks directly> to the fact that identity is a topologically complex matter> and that the trichotomy of the real numbers cannot be taken> for granted.>> >http://blog.sigfpe.com/2008/01/what-does-topology-have-to-do-with.html>> > Our language is countable , the real numbers are not . Thus we don't> > work directly with "the plenum" , the real numbers as infinite strings> > of digits . How could we? Who has time to read an infinite string?>> Well, (in all good humor) Alan Turing for one.>> Correct. While not an application one would have expected for> automata, the fact is that a representation of the logical problem> using automata led to reduced machines because of the Euclidean> algorithm.>> > What we do work with are the 'finite definitions' of these 'infinite> > numbers' , for all the real numbers we can think about .Whether any> > other kind of number is "real" ,other than those we can think about,> > depends on your "orientation" in mathematics , though I affirm they're> > not 'real' .> > Now , these 'finite definitions' "subdue" the infinite numbers, making> > their contents accessible to our tiny, finite minds . Thus , equality> > becomes decidable , and 1 = 0.(9) while 1 not = 0.9999998(9) .>> Thanks to my recent postings on sci.logic and sci.math, I> define "subdue" to be the algebraic extension field over the> rationals characterized by>> a+b*surd(5)>> > One question remains : Is anything lost when "replacing" these> > "infinite objects" by their finite definitions?>> No, but something is lost when we treat>> 1.000... = 0.999...>> trivially if, in fact, one is interested in how one uses> definition and the sign of equality to 'subdue'.>> That was an important point of the post. The decision turns> a lossless representation into a lossy representation.>> What branch of mathematics other than finite-state automata> could even represent that?>> > The beautiful fact is that the objects of mathematics are analytic ,> > not synthetic , thus nothing is lost in terms of meaning by saying> > 0.(9) instead of 0.99999.... and much is gained in terms of what we> > could do with "the finite 0.(9)" , as opposed to "the infinite> > 0.9999..." .>> The analytic/synthetic distinction? Are you sure you> want to go there?

The problem of 1 = 0.(x) appears for any possible base ofnumeration .If you're bothered by the representation being 'lossy' , you canalways try continued fractions for the numbers in the interval [0,1] :Each real number is represented by a (possibly infinite) sequence ofstrictly positive integers :You represent r by [a1,a2,a3 ..... an ...] meaning thatr = 0 + 1/ (a1 + 1 / (a2 + 1/ (a3 + .... )))I'm pretty sure you can build up the whole of analysis this way ,though nobody's bothered to do it, so it must be tedious.That being said, I was never really bothered by the whole 0.(9) = 1business , it's just a quirk in notation .What seems far more troublesome is the representation of finitefields , you always have to 'choose' one of many irreduciblepolynomials if you want to work with them .